A mean field model for the interactions between firms on the markets of their inputs

TL;DR

This paper develops a mean field model to analyze long-term interactions among heterogeneous firms in input markets, using coupled differential equations to describe equilibrium states.

Contribution

It introduces a novel mean field framework combining Hamilton-Jacobi and continuity equations for modeling firm interactions and proves the existence of equilibria.

Findings

Existence of equilibrium states under certain assumptions.

Coupled non-linear differential equations effectively model firm interactions.

Framework captures heterogeneity and market dynamics of firms.

Abstract

We consider an economy made of competing firms which are heterogeneous in their capital and use several inputs for producing goods. Their consumption policy is fixed rationally by maximizing a utility and their capital cannot fall below a given threshold (state constraint). We aim at modeling the interactions between firms on the markets of the different inputs on the long term. The stationary equlibria are described by a system of coupled non-linear differential equations: a Hamilton-Jacobi equation describing the optimal control problem of a single atomistic firm; a continuity equation describing the distribution of the individual state variable (the capital) in the population of firms; the equilibria on the markets of the production factors. We prove the existence of equilibria under suitable assumptions.